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# Bhāskara I

Bhāskara (commonly called Bhāskara I to avoid confusion with the 12th century mathematician Bhāskara II) (c. 600 - c. 680) was a 7th century Indian mathematician, who was apparently the first to write numbers in the Hindu-Arabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work.

## Biography

We know little about Bhāskara's life. Presumably he was born in Kerala as a Nambudiri Brahmin. His astronomical education was given by his father. Bhaskara is considered the most important scholar of Aryabhata's astronomical school.He and Brahmagupta are the most renowed Indian mathematicians who made considerable contributions to the study of fractions

## Representation of numbers

Bhaskara's probably most important mathematical contribution concerns the representation of numbers in a positional system. The first positional representations were known to Indian astronomers about 500. However, the numbers were not written in figures, but in words or allegories, and were organized in verses. For instance, the number 1 was given as moon, since it exists only once; the number 2 was represented by wings, twins, or eyes, since they always occur in pairs; the number 5 was given by the (5) senses. Similar to our current decimal system, these words were aligned such that each number assigns the factor of the power of ten corresponding to its position, only in reverse order: the higher powers were right from the lower ones. For example,

1052 = wings senses void moon.

Why did the Indian scientists use words instead of the already known Brahmi numerals? The texts were written in Sanskrit, the "language of the gods", which played a similar role as Latin in Europe, the spoken languages were quite different dialects. Presumably, the Brahmi numerals which were used in every-day life were regarded as too vulgar for the gods (Ifrah 2000, p. 431).

About 510, Aryabhata used a different method ("Aryabhata cipher") assigning syllables to the numbers. His number system has the basis 100, and not 10 (Ifrah 2000, p. 449). In his commentary to Aryabhata's Aryabhatiya in 629, Bhaskara modified this system to a true positional system with the base 10, containing a zero. He used properly defined words for the numbers, began with the ones, then writes the tens, etc. For instance, he wrote the number 4,320,000 as

viyat ambara akasha sunya yama rama veda
sky atmosphere ether void primordial couple (Yama & Yami) Rama Veda
0 0 0 0 2 3 4

His system is truly positional, since the same words representing, e.g., the number 4 (like veda), can also be used to represent the values 40 or 400 (van der Waerden 1966, p. 90). Quite remarkably, he often explains a number given in this system, using the formula ankair api ("in figures this reads"), by repeating it written with the first nine Brahmi numerals, using a small circle for the zero (Ifrah 2000, p. 415). Contrary to his word number system, however, the figures are written in descending valuedness from left to right, exactly as we do it today. Therefore, at least since 629 the decimal system is definitely known to the Indian scientists. Presumably, Bhaskara did not invent it, but he was the first having no compunctions to use the Brahmi numerals in a scientific contribution in Sanskrit.

The first, however, to compute with the zero as a number and to know negative numbers, was Bhaskara's contemporary Brahmagupta.

## Further contributions

Bhaskara wrote three astronomical contributions. In 629 he commented the Aryabhatiya, written in verses, about mathematical astronomy. The comments referred exactly to the 33 verses dealing with mathematics. There he considered variable equations and trigonometric formulas.

His work Mahabhaskariya divides into eight chapters about mathematical astronomy. In chapter 7, he gives a remarkable approximation formula for $sin x$, that is

$sin x approx frac\left\{16x \left(pi - x\right)\right\}\left\{5 pi^2 - 4x \left(pi - x\right)\right\}, qquad \left(0 leq x leq frac\left\{pi\right\}\left\{2\right\} \right)$
which he assigns to Aryabhata. It reveals a relative error of less than 1.9% (the greatest deviation $frac\left\{16\right\}\left\{5pi\right\} - 1 approx 1.859%$ at $x=0$). Moreover, relations between sine and cosine, as well as between the sine of an angle $>90^circ$, $>180^circ$ or $>270^circ$ to the sine of an angle $<90^circ$ are given. Parts of Mahabhaskariya were later translated into Arabic.

Bhaskara already dealt with the assertion: If $p$ is a prime number, then $1 + \left(p-1\right)!$ is divisible by $p$. It was proved later by Al-Haitham, also mentioned by Fibonacci, and is now known as Wilson's theorem.

Moreover, Bhaskara stated theorems about the solutions of today so called Pell equations. For instance, he posed the problem: "Tell me, O mathematician, what is that square which multiplied by 8 becomes - together with unity - a square?" In modern notation, he asked for the solutions of the Pell equation $8x^2 + 1 = y^2$. It has the simple solution $x = 1$, $y = 3$, or shortly $\left(x,y\right) = \left(1,3\right)$, from which further solutions can be constructed, e.g., $\left(x,y\right) = \left(6,17\right)$.

## References

• H.-W. Alten, A. Djafari Naini, M. Folkerts, H. Schlosser, K.-H. Schlote, H. Wußing: 4000 Jahre Algebra. Springer-Verlag Berlin Heidelberg 2003 [ISBN 3-540-43554-9], §3.2.1
• S. Gottwald, H.-J. Ilgauds, K.-H. Schlote (Hrsg.): Lexikon bedeutender Mathematiker. Verlag Harri Thun, Frankfurt a. M. 1990 [ISBN 3-8171-1164-9]
• G. Ifrah: The Universal History of Numbers. John Wiley & Sons, New York 2000 [ISBN 0-471-39340-1]
• B. van der Waerden: Erwachende Wissenschaft. Ägyptische, babylonische und griechische Mathematik. Birkäuser-Verlag Basel Stuttgart 1966